We study a Sturm-Liouville type eigenvalue problem for second-order differential equations on the infinite interval (-∞, ∞).Here the eigenfunctions are nonzero solutions exponentially decaying at infinity.We prove that at any discrete eigenvalue the differential equations are integrable in the setting of differential Galois theory under general assumptions.Our result is illustrated with three examples for a stationary Schrödinger equation having a generalized Hulthén potential; a linear stability equation for a traveling front in the Allen-Cahn equation; and an eigenvalue problem related to the Lamé equation.